Divisor Calculator

Find the prime factorization $n = p_1^{a_1} \times p_2^{a_2} \times \ldots$, then compute $d(n) = (a_1 + 1)(a_2 + 1)\ldots$. For example, $60 = 2^2 \times 3 \times 5$ has $d(60) = (2+1)(1+1)(1+1) = 12$ divisors.

Use the formula $\sigma(n) = \prod (p_i^{a_i+1} - 1)/(p_i - 1)$ for each prime power in the factorization. For $60 = 2^2 \times 3 \times 5$, $\sigma(60) = (2^3 - 1)/(2 - 1) \times (3^2 - 1)/(3 - 1) \times (5^2 - 1)/(5 - 1) = 7 \times 4 \times 6 = 168$.

A perfect number is a positive integer that equals the sum of its proper divisors, which means $\sigma(n) = 2n$. The first three perfect numbers are 6 (= 1+2+3), 28 (= 1+2+4+7+14), and 496. By the Euclid-Euler theorem, every even perfect number has the form $2^{p-1}(2^p - 1)$ where $2^p - 1$ is a Mersenne prime.

Highly composite numbers have more divisors than any smaller positive integer. They tend to be products of the smallest primes with descending exponents, like $12 = 2^2 \times 3$ (6 divisors), $60 = 2^2 \times 3 \times 5$ (12 divisors), and $120 = 2^3 \times 3 \times 5$ (16 divisors).

The divisor function $d(n)$ is multiplicative, meaning $d(a \times b) = d(a) \times d(b)$ whenever $\gcd(a, b) = 1$. This follows from the fact that each divisor of $a \times b$ uniquely corresponds to a pair of divisors, one from $a$ and one from $b$, when $a$ and $b$ share no common prime factor.